Children are one of the most under-represented groups in speech technologies, as well as one of the most vulnerable in terms of privacy. Despite this, anonymization techniques targeting this population have received little attention. In this study, we seek to bridge this gap, and establish a baseline for the use of voice anonymization techniques designed for adult speech when applied to children's voices. Such an evaluation is essential, as children's speech presents a distinct set of challenges when compared to that of adults. This study comprises three children's datasets, six anonymization methods, and objective and subjective utility metrics for evaluation. Our results show that existing systems for adults are still able to protect children's voice privacy, but suffer from much higher utility degradation. In addition, our subjective study displays the challenges of automatic evaluation methods for speech quality in children's speech, highlighting the need for further research.